We have differential equation,
$\frac{dy}{dt} = ky$
where, $y$ is quantity that increases or decreases at a rate that at any given time $t$ is proportional to the amount present.
In order to derive basic equation of growth and decay from differential equation of proportional change, we follow this.
Now to derive formal equation for law of exponential change, we follow this.
Thus, the law of exponential change is given by,
$y = y_0 e^{kt}$.
If $k > 0$, it is Growth and if $k < 0$, it is Decay.
References:
- Book Calculus by Thomas/Finney 9 Th Edition, Page 482
- “The Law of Exponential Change -Growth and Decay.” [Online]. Available: https://content.dodea.edu/VS/HS/AP_Calculus/websites/AP_Calculus_M6/section6_4/docs/Lesson%2006.04.01%20The%20Law%20of%20Exponential%20Change.pdf